Pfeifer records process

Abstract

In this article, we study the limit distribution of sums of Pfeifer records. Motivated by the results obtained by Arnold and Villaseñor [Generalized order statistics process and Pfeifer records, Statistics 46(3) (2012), pp. 373–385], we show that the partial sum process of Pfeifer records converge to a function of the Brownian motion. The normalization is either a sequence of appropriate constants or a sequence of functions, depending on the tail behaviour of the underlying variables. These results, in particular, prove stronger version of results obtained in Villaseñor and Arnold [On limit laws for sums of Pfeifer records, Extremes 10 (2007), pp. 235–248] and Bose and Gangopadhyay [Convergence of linear functions of Pfeifer records, Extremes 13 (2010), pp. 89–106] and extends results of Bose et al. [Partial sum process for records, Extremes 8 (2005), pp. 43–56] from classical records to Pfeifer records.

Keywords:

• Brownian motion
• Pfeifer records
• record values
• regularly varying function
• rapidly varying function
• limiting process

AMS 2000 Subject Classifications:

• 60F05
• 60F15
• 60G70

Acknowledgements

We thank the referee for careful reading of the manuscript. Arup Bose research was supported by J.C. Bose National Fellowship, Department of Science and Technology, Government of India. Part of the work was done while visiting Department of Economics, University of Cincinnati, USA.